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Full text of "Treatise On Analysis Vol-Ii"

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We begin by showing that the topology of E is Hausdorff. Let *, y be two
distinct points of E. If there exists an index n such that x e Un and y $ ~Cn ,
then V = Uw and W = E - Ow are open neighborhoods of x and *y respectively
which do not intersect. If on the other hand x e UM and y e Un for some n,
then in the metrizable subspace Ort there exists an open neighborhood V0 of x
and an open neighborhood W0 of y which do not intersect. The set V =
V0 n Un is an open neighborhood of x in E, and there exists an open set W in
E such that W n 0M = W0 . Hence W is an open neighborhood of y in E, and
V n W = 0.

For each n, let dn be a distance defining the topology of Ort , and let
(VaJwgji be a basis for the topology of Un, where Vm/J is an open ball with
center amn and radius rmn (3.9.4). Letfmn be the function which is equal to
rmn - dn(amny x) in VWII and is 0 in the complement of Vmrt in E. Then/mn is
continuous on E, since it vanishes at the frontier points of VW/J . Now let

for all x, y in E. By virtue of (12.4.6), the proof will be complete if we show
that the pseudo-distances dnm define the topology of E.

For each x0 e E, every set defined by an inequality of the form
dmn(x0 ,x)< a is open in E (3.11.4). Conversely, there exists an integer n such
that x0 E Un , and for each neighborhood W of *0 in E there exists an integer
W such that Vmn is an open neighborhood of x0 contained in W (3.9.3).
Hence fmn(x0) = ft > 0, and the set of all x e E such that dmn(x0 , x) < %fl is
therefore contained in W. By virtue of (12.2.1), this completes the proof.

We remark that the conclusion of (12.4.7) is not valid without the hypothe-
sis of denumerability on the open covering (UA) (Section 12.16, Problem 22);
again, the hypotheses cannot be weakened by assuming merely that the open
sets Urt are separable and metrizable (section 12.3, Problem 6(f)).


1.    Let E be a topological space such that, for each x e E, the neighborhoods of x which are
both open and closed in E form a fundamental system of neighborhoods of x. Show that
E is uniformizable. (Observe that the characteristic function of a set which is both open
and closed in E is continuous on E.)

2.    Let E be a denumerable Hausdorff uniformizable space. Show that, for each x e E,
the open-and-closed neighborhoods of x form* a fundamental system of neighborhoods



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